Given, equation of circle ‌x2+y2−4x−6y−12=0....(i) ‌‌ centre ‌(−g,−f)=(2,3) ‌‌ And radius ‌=√g2+f2−c ‌‌‌=√4+9+12=√25=5
Equation of new circle whose radius is 3 ‌(x−h)2+(y−k)2=(3)2 ‌⇒‌‌(x−h)2+(y−k)2=9....(ii) Since circle (i) and (ii) touches each other internally at (−1,−1) ∴‌‌C1C2=|r1−r2| where c2=(h,k) ∴‌‌C1C2=|5−3|=2 P divides the line segment C1C2 in the ratio r1:r2 ∴C1P‌=r1‌ and ‌C2P=r2 C1P‌=√(−1−2)2+(−1−3)2 ‌=√(−3)2+(−4)2 ‌=√9+16=√25=5 and C2P=r2=3 ∴ The point of tangency P divides the line segment C1C2 externally in the ratio r1:−r ∴‌P=‌